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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="yellow_0.html#V2">upper-bounded</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<a NAME="E1:14"/><i><font color="Green">E7</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E2:14"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E2:14_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:14_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:14_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="orders_4.html#E1:14_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:14_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:14_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="orders_4.html#E4:14_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="orders_4.html#R1" target="_self">form_upper_lower_partition_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="orders_4.html#E1:14"><i><font color="Green">E7</font></i></a>, <a class="ref" href="orders_4.html#D3" target="_self">Def3</a></i>;<br/>


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